I would like to apply the Bernstein-von Mises theorem in a very simple settings to be able to make the following claim:
Let $\theta \in [0,1]$ be a random parameter with strictly positive and bounded probability density function. Its observations $X_1,X_2,\ldots$ are i.i.d. normally distributed, specifically $X_n \sim N(\theta, 1)$. Then $\sqrt{n}(\theta-\theta_0)$ converges in distribution to $N(0,1)$.
What are the regularity conditions that need to be satisfied? Wikipedia page on the Bernstein-von Mises theorem only states "finite-dimensional, well-specified, smooth, existence of tests". However, I would need a precise formulation of the assumption for this simple one-dimensional setting. What would be a good reference book with a simple formulation of the theorem? I found only the book Asymptotic Statistics by A. W. van der Vaart, which however treats unnecessarily general case.